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Luttinger–Ward Functional
In solid state physics, the Luttinger–Ward functional, proposed by Joaquin Mazdak Luttinger and John Clive Ward in 1960, is a scalar functional of the bare electron-electron interaction and the renormalized one-particle propagator. In terms of Feynman diagrams, the Luttinger–Ward functional is the sum of all closed, bold, two-particle irreducible diagrams, i.e., all diagrams without particles going in or out that do not fall apart if one removes two propagator lines. It is usually written as \Phi /math> or \Phi ,U/math>, where G is the one-particle Green's function and U is the bare interaction. The Luttinger–Ward functional has no direct physical meaning, but it is useful in proving conservation laws. The functional is closely related to the Baym–Kadanoff functional constructed independently by Gordon Baym and Leo Kadanoff in 1961. Some authors use the terms interchangeably; if a distinction is made, then the Baym–Kadanoff functional is identical to the two-particle i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid State Physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Background Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern ( crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Functional
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phys
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Review
''Physical Review'' is a peer-reviewed scientific journal established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the American Physical Society (APS). The journal is in its third series, and is split in several sub-journals each covering a particular field of physics. It has a sister journal, ''Physical Review Letters'', which publishes shorter articles of broader interest. History ''Physical Review'' commenced publication in July 1893, organized by Cornell University professor Edward Nichols and helped by the new president of Cornell, J. Gould Schurman. The journal was managed and edited at Cornell in upstate New York from 1893 to 1913 by Nichols, Ernest Merritt, and Frederick Bedell. The 33 volumes published during this time constitute ''Physical Review Series I''. The American Physical Society (APS), founded in 1899, took over its publication in 1913 and star ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ward Identity
Ward may refer to: Division or unit * Hospital ward, a hospital division, floor, or room set aside for a particular class or group of patients, for example the psychiatric ward * Prison ward, a division of a penal institution such as a prison * Ward (electoral subdivision), electoral district or unit of local government ** Ward (KPK), local government in Khyber Pakhtunkhwa, Pakistan ** Ward (South Africa) ** Wards of Bangladesh ** Wards of Germany ** Wards of Japan ** Wards of Myanmar ** Wards and electoral divisions of the United Kingdom ** Ward (United States) *** Wards of New Orleans * Ward (fortification), part of a castle * Ward (LDS Church), a local congregation of The Church of Jesus Christ of Latter-day Saints * Ward (Vietnam), a type of third-tier subdivision of Vietnam Entertainment, arts and media * WOUF (AM), a radio station (750 AM) licensed to serve Petoskey, Michigan, United States, which held the call sign WARD from 2008 to 2021 * Ward Cleaver, a fictional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luttinger's Theorem
In condensed matter physics, Luttinger's theorem is a result derived by J. M. Luttinger and J. C. Ward in 1960 that has broad implications in the field of electron transport. It arises frequently in theoretical models of correlated electrons, such as the high-temperature superconductors, and in photoemission, where a metal's Fermi surface can be directly observed. Definition Luttinger's theorem states that the volume enclosed by a material's Fermi surface is directly proportional to the particle density. While the theorem is an immediate result of the Pauli exclusion principle in the case of noninteracting particles, it remains true even as interactions between particles are taken into consideration provided that the appropriate definitions of Fermi surface and particle density are adopted. Specifically, in the interacting case the Fermi surface must be defined according to the criteria that :G(\omega=0,\,p) \to 0 or \infty, where G is the single-particle Green function in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Mean Field Theory
Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in density functional theory and usual band structure calculations, breaks down. Dynamical mean-field theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the nearly free electron gas limit and the atomic limit of condensed-matter physics. DMFT consists in mapping a many-body lattice problem to a many-body ''local'' problem, called an impurity model. While the lattice problem is in general intractable, the impurity model is usually solvable through various schemes. The mapping in itself does not constitute an approximation. The only approximation made in ordinary DMFT schemes is to assume the lattice self-energy to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GW Approximation
The ''GW'' approximation (GWA) is an approximation made in order to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the self-energy ''Σ'' in terms of the single particle Green's function ''G'' and the screened Coulomb interaction ''W'' (in units of \hbar=1) : \Sigma = iGW - GWGWG + \cdots can be truncated after the first term: : \Sigma \approx iG W In other words, the self-energy is expanded in a formal Taylor series in powers of the screened interaction ''W'' and the lowest order term is kept in the expansion in GWA. Theory The above formulae are schematic in nature and show the overall idea of the approximation. More precisely, if we label an electron coordinate with its position, spin, and time and bundle all three into a composite index (the numbers 1, 2, etc.), we have : \Sigma(1,2) = iG(1,2)W(1^+,2) - \int d3 \int d4 \, G(1,3)G(3,4)G(4,2)W(1,4)W(3,2) + ... where the "+" superscript means the time index is shif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hubbard Model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each electron experiences competing forces: one pushes it to tunnel to neighboring atoms, while the other pushes it away from its neighbors. Its Hamiltonian thus has two terms: a kinetic term allowing for tunneling ("hopping") of particles between lattice sites and a potential term reflecting on-site interaction. The particles can either be fermions, as in Hubbard's original work, or bosons, in which case the model is referred to as the "Bose–Hubbard model". The Hubbard model is a useful approximation for particles in a periodic potential at sufficiently low temperatures, where all the particles may be assumed to be in the lowest Bloch band, and long-range interactions between the particles can be ignored. If interactions between particles at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Potential
The grand potential is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble. Definition Grand potential is defined by : \Phi_ \ \stackrel\ U - T S - \mu N where ''U'' is the internal energy, ''T'' is the temperature of the system, ''S'' is the entropy, μ is the chemical potential, and ''N'' is the number of particles in the system. The change in the grand potential is given by : \begin d\Phi_ & = dU - TdS - SdT - \mu dN - Nd\mu \\ & = - P dV - S dT - N d\mu \end where ''P'' is pressure and ''V'' is volume, using the fundamental thermodynamic relation (combined first and second thermodynamic laws); :dU = TdS - PdV + \mu dN When the system is in thermodynamic equilibrium, ΦG is a minimum. This can be seen by considering that dΦG is zero if the volume is fixed and the temperature and chemical potential have stopped evolving. Landau free ene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagrammatic Expansion Of The Luttinger–Ward Functional
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word ''graph'' is sometimes used as a synonym for diagram. Overview The term "diagram" in its commonly used sense can have a general or specific meaning: * ''visual information device'' : Like the term "illustration", "diagram" is used as a collective term standing for the whole class of technical genres, including graphs, technical drawings and tables. * ''specific kind of visual display'' : This is the genre that shows qualitative data with shapes that are connected by lines, arrows, or other visual links. In science the term is used in both ways. For example, Anderson (1997) stated more generally: "diagrams are pictorial, yet abstract, representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-energy
In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between the particle and its environment. In electrostatics, the energy required to assemble the charge distribution takes the form of self-energy by bringing in the constituent charges from infinity, where the electric force goes to zero. In a condensed matter context relevant to electrons moving in a material, the self-energy represents the potential felt by the electron due to the surrounding medium's interactions with it. Since electrons repel each other the moving electron polarizes, or causes to displace the electrons in its vicinity and then changes the potential of the moving electron fields. These are examples of self-energy. Characteristics Mathematically, this energy is equal to the so-called on mass shell value of the proper self- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
